NASA CONTRACTOR REPORT AN EXPLORATORY STUDY OF INTERNALLY AND EXTERNALLY FINNED RADIATORS FOR BRAYTON CYCLES IN SPACE by S. V. Mmzson Prepared by S. V. MANSON & COMPANY, INC. Arlington, Va. for Lewis Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . WASHINGTON, D. C. . MARCH 1967 ; \ ik 1. ;,< mm-- mm..------.-. -II.---,.. .I mm-1 -- .--1.-.1...111..1... 1111.-m https://ntrs.nasa.gov/search.jsp?R=19670012119 2018-05-30T03:18:47+00:00Z
Transcript
NASA CONTRACTOR
REPORT
AN EXPLORATORY STUDY OF INTERNALLY AND EXTERNALLY FINNED RADIATORS FOR BRAYTON CYCLES IN SPACE
by S. V. Mmzson
Prepared by
S. V. MANSON & COMPANY, INC.
Arlington, Va.
for Lewis Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . WASHINGTON, D. C. . MARCH 1967
Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.
Prepared under Contract No. NAS 3-2535 by S. V. MANSON & COMPANY, INC.
Arlington, Va.
for Lewis Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sole by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151 - CFSTI price $3.00
FOREWORD
The research described herein was conducted by S. V. Manson & Com-
pany, Inc., under NASA Contract No. NAS 3-2535. Mr. Martin Gutstein,
Space Power Systems Division, NASA Lewis Research Center, was Tech-
AN EXPLORATORY STUDY OF INTERNALLY AND EXTERNALLY FINNED
RADIATORS FOR BRAYTON CYCLES IN SPACE
S. V. Manson
SUMMARY
Sizes and weights are computed for Brayton cycle radiators that use
a gas as their working fluid. The effects of fins on the Inside
surfaces of the radiator tubes are evaluated. The effects of annu-
lar fins on the outside surfaces of the radiator tubes are discussed.
The calculations Indicate that Internally finned radiators are more
than 15 percent lighter in weight and more than 35 percent smaller
in size than are Internally unflnned radiators.
The calculations suggest that radiators equipped with annular exter-
nal fins may be smaller in size than radiators equipped with central-
type external fins.
INTRODUCTION
The Brayton cycle, which employs a gas as the turbomachinery work-
ing fluid, is one of the thermodynamic cycles being considered for
the conversion of heat to electrical power In space applications
(Ref. 1). Two maJor reasons for considering the Brayton cycle are
(1) that there exists a large background of successful experience
2
with gas cycle turbomachinery and (2) that the use of a gas avoids
fluid flow, heat transfer, component and materials problems that
may require solution with two-phase fluids in space.
One possible arrangement of the Brayton cycle is shown schematically
in figure 1. In this arrangement the working gas goes through the
following processes:
(a) It is heated In the heat source;
(b) Flows to the turbine, where It expands and delivers to the tur-
bine the energy required to drive the compressor and alternator;
(c) Flows to the recuperator, where it transfers heat to a relatively
cool portion of the cycle gas stream;
(d) Flows to the radiator, where it discards the waste heat of the
cycle;
(e) Flows to the compressor, where its pressure and temperature are
raised;
(f) Flows to the recuperator, where it is heated;
(g) Flows to the heat source, where It is heated further (Step (a)).
The gas goes through Steps (a)-(g) repeatedly. For the present
study, Step (d) is of primary interest; this Step indicates that the
working fluid in the radiator is a gas.
When the working fluid in the waste heat radiator Is a gas, the ra-
diator size, weight and reliability are affected by the following
properties of the gas: (1) Gases are relatively poor heat transfer
fluids; (2) gases have low densities and require relatively large
3
flow areas i'n order to avoid high pressure drop and substantial
pumping power; (3) g ases experience a temperature drop during flow
through the radiator with an associated decrease in the radiating
potential of the armor and fins.
The cited gas properties could lead to large, heavy and thermally
stressed waste heat radiators.
The aims of the present study are as follows:
(a) To develop a method of computing the dimensions and weights of
radiators that use a gas as their working fluid;
(b) To employ this method to evaluate two concepts for reducing size,
weight and stress In such radiators. The first concept involves
the use of conducting fins on the gas-swept inner surfaces of
the radiator tubes. The second concept involves the use of
annular radiating fins on the external surfaces of the radiator
tubes. The effects on radiator size and weight are evaluated
quantitatively. The effects on thermal stress are discussed
qualitatively.
The general radiator arrangement within which the foregoing concepts
are evaluated Is shown In figure 2. The radiator consists of an
assembly of tubes lying in a single plane and radiating heat to space
on both sides of the plane. Gas is fed to the tubes by a supply
header and is removed from the tubes by an exhaust header; both
headers are tapered. Within the tubes the gas transfers heat by
4
convection to the tube inner surfaces. The heat then flows by con-
duction across the tube walls, which are thick enough to serve as
armor against penetration by meteoroids. Part of the heat is radiated
to space by the outer surface of the armor; the rest of the heat Is
conducted to external fins attached to the outer surface of the armor
and is radiated to space by these fins.
In the present study, the radiator tubes of figure 2 are assumed to
be finned internally, as well as externally. The tube Internal
geometries evaluated are shown in cross section In figure 3.
Figure 3 shows four internally finned tube geometries, and also the
internally bare (unfinned) tube geometry that was computed for
reference purposes. The external fin configurations evaluated are
shown in figures 4 and 5. Figure 4 illustrates conventional central-
type external fins; figure 5 illustrates circumferential (annular)-
type external fins.
In relation to the configurations shown In figures 2 - 5, the alms
of this study may be stated in detail as follows:
1. To develop a method of computing the sizes and weights of
armored, externally finned, headered radiators that are arranged
as In figure 2 and that operate non-isothermally with a gaseous
working fluid.
2. For a prescribed set of operating conditions, and for tubes
equipped with central external fins, to compute the sizes and weights
5
of both Internally finned and Internally unflnned radiators.
3. To compare the sizes and weights of the internally finned
radiators with the sizes and weights of the internally unfinned rad-
iators.
4. To consider briefly the potential gains from use of external
radiating fins of annular shape.
In the calculation procedure developed, the radiator parts are com-
puted in a definite sequence, as follows: (1) armor, (2) external
fins, (3) headers, (4) radiator size and weight.
The armor details are computed with "mechanical" (I.e., non-thermal)
equations. The external fins are computed by use of basic fin-and-
tube data of the sort available In References 2 and 3. The headers
are computed on the basis of gas velocity and pressure drop consider-
ations. Heat transfer from the headers is neglected; the outside
surface area of the headers is calculated, however, and is used as
a basis for estimating the final thickness of the armor on the tubes
and headers of the radiator.
The calculation procedure is applicable to armored-tube radiators
with a wide variety of external fin geometries. For ease of cal-
culation, the present study Is limited to the special class of
radiators for which the ratio of the heat dissipated by the external
fins to the heat dissipated by the armor is the same at every axial
station.
6
Radiators composed entirely of aluminum are assumed. Tube Inside
diameters ranging from about 0.3 to about 3.4 inches, and tube
lengths of 6 and 25 feet, are evaluated. In each of the internally
finned tubes the number of fins per tube is varied over a substan-
tial PUlgej the total range covered for the various internally
finned geometries Is 4 to 70 fins per tube. In all cases the
assumed thickness of the Internal fin metal is .004 Inch. For the
class of radiators considered, the thickness of the external fins
decreases in the direction of the gas flow If the fin length (or
diameter) and fin conductance parameter are both kept constant for
the entire radiator. Constant fin length (or diameter) and constant
fin conductance parameter are assumed in the present report; several
values of the conductance parameter are considered for each of the
two external fin types evaluated.
APPROACH
To calculate radiators of the type illustrated in figure 2, the
approach used in this study is divide the radiator into several
parts and to compute each part separately in a definite sequence.
The sequence is chosen so as to permit the calculation of each part
from existing or previously established information. Wherever
possible, use is made of integral relations and end states to design
each component in its entirety, rather than to pursue step-by-step
calculation procedures.
c- --
7
The parts into which the radiator is divided are as follows:
(a) the internal fins (if any); (b) the armored tubes through
which the gas flows; (c) the external fins'; (d) the headers. The
parts are calculated in the order listed. Qualitative descriptions
of the procedures employed are as follows:
The geometric arrangement and the detailed dimensions of the
internal fins are treated as input.
The tubular armor is treated in part as input, and in part as
output. The tube inside diameter and the tube length are assigned;
the number of tubes and the armor thickness are computed. The
equations employed to compute the number of tubes and the armor
thickness are (1) the gas pressure drop equation, (2) the one-
dimensional gas continuity equation, (3) the equation that defines
the armor thickness for a prescribed degree of protection against
penetration by meteoroids, and (4) a purely geometric equation
that relates the armor surface to the number of armored tubes,
their length, their inside diameter and wall thickness. The wall
thickness computed in this way corresponds, from the viewpoint
of meteoroid protection, to the exposed outer surface of the tubes
alone; a correction to the wall thickness is added later, when the
exposed surface of the headers has been computed.
With the armor geometry known except for a refinement of Its
thickness, the calculation proceeds to the external fins. Part
of the information needed to fix the external fin dimensions is
obtained by introducing the thermal equations of the armor. The
armor thermal equations, together with independently available
8
external fin data, are employed to determine the dimensions of.
the external fins required for thermal compatibility with the
already computed armor. In the present study the armor equations and the external fin data are formulated in terms of a heat ratio,
namely, the ratio of the heat radiated jointly by the armor and the
external fins, to the heat that would be radiated at the same temp-
erature by the armor alone if the external fins were absent. This ratio, which in the general case would vary in magnitude from one
point to another along the armor of a non-isothermal radiator, Is
denoted by the symbol (dQ)/(dQE) . (All symbols are defined in
Appendix A.)
For simplicity, the calculations In this report are confined to
the class of radiators for which the ratio (dQ)/(dQ{) is a constant
for the entire radiator. The basic data of References 2 and 3,
expressed in terms of (dQ)/(dQE), are employed to determine the
dimensions of central and circumferential types of external fins
that are compatible with the armor geometries of this class
of radiators. Graphical maps are used to facilitate the calcul-
ations. Axial temperature variation is taken Into account.
With both the armor geometry and external fin dimensions known, the
associated header lengths are readily computed. The headers are
designed to provide the same bulk fluid velocity at all axial sta-
tions; hence, the headers are tapered along their length. The
entrance and exit diameters are determined by the requirement
that the fluid pressure drop shall be a prescribed value. The
header surface areas are also computed and are used as a basis
9
for estimating the final value of the armor thickness on the
radiator tubes and headers. Heat transfer from the header surfaces
is not taken into account in the calculation procedure.
With all details known, the radiator total size and weight are
computed straightforwardly. The sizes and weights of internally
finned radiators are then compared with the sizes and weights
of internally unfinned radiators. Similarly, the sizes and weights
of radiators equipped with circumferential external fins are
compared with the sizes and weights of radiators eqmipped with
central external fins.
CALCULATION PROCEDURE
The equations of the calculation procedure are indicated In this
section. In addition, the input and output quantities of the
calculation and the major underlying assumptions are indicated.
10
The input consists of the following items (symbols are defined in
Appendix A):
Gas ouerating conditions: Gas composition and r.i~; and Ten, Tex,
Pen> pex during flow through each of the following physical
components-- the supply header, the radiator tubes and the exhaust
header; also, data on cp, IJ- and k as functions of T.
Internal fins (see figure 3): Material, geometric array, b,
Q, n; correlations for heat transfer coefficients and friction
factors in flow through channels containing such fins (see figure 6);
and a formula or curve that permits evaluation of the fin effective-
ness as a function of the parameter(s) on which the fin effectiveness
depends.
Armor: Material, c, di, 2. Also input is a meteoroid criterion
that permits calculation of armor thickness for prescribed values of
P(O) and z; the quantities P(0) and z are input values. In the
Present study the meteoroid criterion of Reference 4 is employed.
External fins: Material, C, general arrangement. In addition,
for central-type external fins the input includes the conductance
parameter (NC L 'F
> and curves of LF/R, versus (dQ)/(dQi) at various
values of NC L (see figure 7). For circumferential external fins, 9F
the input includes Ro,F/Ra and curves of (dQ)/(dQz) versus NC R at 9a
various values of sF/Ra (see figure 8).
II
11
Headers: Gas operating conditions, wall composition, and specific-
ation whether the headers are unsplit or split (see figure 2).
E;Ilvirom : Te
The following items are end results of the calculation:
Armor: N, 6,, Da, weight.
Internal fins: weight.
External fins: For central-type fins -- LF , nF,x, weight. For
circumferential type fins -- Ri F(=Ra) 3 Ro,F 3 SF 3 nF,x, number 9
of fins, weight.
Headerz: $9 dR,en, dR,x, dR,ex, weight. These quantities are
obtained for both the supply header and the exhaust header.
Total radiator: Weight, planform area (including the incremental
area contributed by the headers).
12
The following assumptions are made in order to facilitate analysis
and calculations. Brief discussion of some of the assumptions is
presented.
Assumption 1. In computing the friction pressure drop of the gas in
a radiator channel, an average gas density may be used for the
entire channel. This average density is assumed to be computable
by use of the perfect gas law in conjunction with an average
pressure and an average temperature given by
pen + pex P av =
2 (1)
(2)
Equation (1) is a reasonable assumption when the overall gas pressure
drop is a moderate fraction ( i 0.1) of the gas inlet pressure and
there are no abrupt pressure changes within the channel. In the present
study, AP/pen M 0*05, the gas velocities are subsonic, and abrupt
pressure changes are not expected.
For this study, equation (2) was simplified by taking CT = 1 l
Check calculations showed that at the radiator operating conditions
considered, the use of CT = 1 over-estimates the radiator sizes and
weights by about 3 to 5 percent.
13
The assumption that an average gas density may be used for the entire
channel makes possible the use of an integral form of the pressure drop
equation. Thereby the assumption uncouples the required number of
radiator tubes from details of the thermal history of the gas. Assump-
tion 1 therefore plays an important role in the calculation procedure
of the present study.
Assumption 2. A single (average) value of the gas heat transfer
coefficient may be used everywhere in the radiator channels. In
computing the average heat transfer coefficient, the physical properties
of the gas may be evaluated at an average gas film temperature given by
, T Tfilm =
i3,av + Tw av 9 2
It is assumed that a satisfactory estimate can be made of Tw av -- If 9
necessary, on the basis of a detailed initial calculation. Preliminary
calculations indicated that TfilmcO.97 Tg,av in the internally
finned radiators of this study. The same value of Tfilm was used
for the internally unfinned radiators of this report.
The assumption that an average gas heat transfer coefficient may be
used for the entire radiator frees the heat transfer coefficient from
the detailed thermal history of the gas. It also implies that a
constant value of sverall coefficient of heat convection-and-conduction,
U, may be employed for the entire radiator. The constancy of U permits
the extraction of U from under an integral sign that arises in a
thermal equation of the armor.
14
Assumption 3. At every tube cross section, the tube wall temperature
is uniform around the circumference.
This assumption facilitates calculations; It permits the use of
numerical data presented in References 2 and 3 for determining the
dimensions of the external fins.
Assumption 4. Axial heat conduction is negligible.
Reference 5 has shown that axial conduction effects are negligible
in the external (radiating) fins of practical radiators, and that
axial temperature variation affects negligibly the radiant heat
interchange between radiator elements. Reference 5 does not study
the effects of axial conduction in the tube wall. A detailed study
of such effects is outside the scope of the present analysis.
Assumption 5. The emissivity and absorptivity of the armor and of
the external fins are uniform over the entire radiator.
Assumption 6. The effective environment temperature is the same for
all parts of the radiator.
Assumption 7. The geometry in the interior of the gas channels is
the same throughout the radiator.
15
Princinal Eauations
Internal Flng
The geometry and dimensions of the internal fins are input data.
Relations involving the internal fins are presented In the first
sub-section under f1Armort8.
Armor
Basic relations for the tube interior:
Aflow per tube = fdi2 - (Blocked area per tube) (4)
Sum of the cross sectional areas of all fin metal parts and of Internal blockage tube (if any), computed at any tube (5) section taken perpendicular to the tube axis; see figure 3.
Sum of the perimeters of those parts of the tube, fins and blockage tube (if
P w per tube = that are contacted by the gas, computed (6) at any tube cross section taken perpen- dicular to the tube axis; see figure 3.
(7)
16
Sum of the wetted perimeters of all fins in any one tube cross section, excluding the fin
Sf and their exposed sides; see figure 9(a).
s = >
(8)
w,i Wetted perimeter of tube wall at the same cross section, including the fin bases and their exposed sides; see figure 9(a).
The definition in equation (8) treats the fin base, which attaches
the fin to the tube wall (see figure 91, as a portion of the tube
wall. This is a close approximation if the fin thickness, 6f, is
very small in comparison with the tube radius, ri, and in comparison
with the fin dimension, b (see figure 9(b) ). In the present study,
the ratio 6f/r. is s 0.01, and the ratio 6/b is 5 0.03. For 1 geometries in which the fins are formed by extrusion and there is
no fin base, the wetted perimeter of the fin base (and of its sides)
is zero, and equation (8) also applies.
For internally unfinned tubes, all terms involving internal fins in
the foregoing equations have the value zero.
The mass flow per unit flow area in the tube is
lil G =
NA flow per tube
The film Reynolds number of the flow is
G deq Tav Refilm = IJ.film T
film
(9)
(10)
For assumed uniform gas flow distribution in the tubes, G and Refilm
are the same for all the tubes,
17
The one-dimensional continuity equation for flow in tubes of constant
cross sectional area Is
G = (pv), = constant (11)
The drop in static pressure experienced by the gas in flow through
the tubes is given by
Apm = + AP momentum
From a generalization of the results of Reference 6 for flow
through tubes,
I I 2 Pfilm v:v
Apfr = 4 ffilm - deq 243
(12)
(13) 1 (pav Tav'T
2 filmlvav
= 4 ffilm d eq a
In equation (131, the friction factor depends on the internal
geometry of the tube and on Refilm, and for each internal geometry
is obtained from a functional relation of the form
ffilm = function of Refilm (see figure 6) (14)
18
Also, from the basic definition of pressure change accompanying
changes in the momentum of a gas during flow through a channel of
constant cross-sectional area,
I- 1
AP momentum = 1 pex:eX)Vex - ("%"") V-J (15)
The gas heat transfer coefficient depends on the internal geometry
of the tube, on the Prandtl number of the fluid and on Refilm, and
for each internal geometry is obtained from a functional relation
of the form
2/3 Prfilm = function of Refilm (see figure 6) (16)
The heat transfer coefficient determined by equation (16) applies
to both the tube surface and fin surface in the tube interior.
For internal fins of the types considered in the present study, the
fin effectiveness is given by the following equation (Ref. 7):
rl, = (17)
I --
19
The total effective heat transfer surface in the tube interiors is
S eff = S W,i + v, Sf
The effective conductance of the gas is
(hS)eff = h(Sw,i + vf Sf)
Equation (19) shows that the effective heat transfer coefficient
relative to the inner surface of the tube walls, (i.e., relative
to %,i 1, 1s
Sf h eff = r)f -
SW,i
(18)
(19)
(20)
For internally unfinned tubes, V, = 0 in equations Cl?)-(20).
The foregoing equations, together with the perfect gas law and
equations (l>-(3) presented in the Assumptions, comprise the basic
elements of calculation insofar as the interior of the gas channels
is concerned.
20
The first steps of the calculation consist of using the input gas
data to compute pav, T, and Tfilm with equations (l)-(3) of the
Assumptions. The bulk average density, Pa,9 Is then computed with
the perfect gas law in conjunction with pay9 T,, and the gas constant
corresponding to the gas composition stipulated in the input. The
pertinent gas properties are then determined on the basis of T,,
and Tfilm (equation (16)). Also computed are the inlet and exit
gas densities, Pen and P,, , for the supply header, the radiator
tubes and the exhaust header.
The Input geometric data for the Internal fins and for the inside
diameter of the armor are then employed in equations (4)-(8) to
'Ompute Aflow per tube 3 deq ' and Sf/Q l
With deq' 'film and Tav/Tfilm known, equation (10) is written in
the forms
Re film = constx G, or, G = c0nst.x Refilm
which permit immediate determination of the value of G associated
with any value of Refilm . Additionally, equations (14) and (16),
used together with data of the sort presented in figure 6, permit
determination of unique values of ffilm and h associated with
each Refilm in channels of prescribed internal geometry.
Calculation sequences based on the foregoing equations are detailed
In the following paragraphs.
21
Number of tub=: The number of tubes is determined by joint use
of the gas continuity and gas pressure drop equations, (11) and
(12)-(151, in conjunction with input data, as follows:
When equation (11) is inserted into equations (13) and (15),
equation (12) takes the familiar form
AP = 4 ffilm 2 G2 T,, + G2 (21)
d eq 2g Pav Tfilm g Pen
In equation (211, the quantPties Ap and Z are input data, and the
gas densities, TaV/Tfilm and deq are known from calculations based
on input data. Thus In equation (211, Only ffilm and G are unknown.
Now the discussion at the end of the foregoing section Indicates
that for known pfilm, Tav/Tfilm and deq in channels of prescribed
internal geometry, both ffilm and G are uniquely determined for each
value of Refilm . Hence, equation (21) is solved for G (and for ffilm)
by iteration of Refilm . Thus G becomes known.
For known G, the number of tubes is computed with equation (9),
re-written in the form
Ii N =
1 A flow per tube
G (22)
in which & is known as input and Aflow per tube is known from previous
calculations ( eq. (4) >. In a final radiator design, N must be an
integer.
I
22
Armor surface and thicknesg: The armor surface and the associated
thickness are determined by joint use of an input meteoroid criterion
and a purely geometric equation for the exposed surface of the tubes:
The armor thickness required to protect the exposed surface of the
radiator tubes Is (Ref. 4)
‘a = ca sa 1/3B (23)
in which Ca and !3 are input constants; C, is given by
In equations (32) and (331, the value of (dQ)/(dQE) is unknown. An
auxiliary relation is required for determining (dQ)/(dQE). Such a
relation is supplied by the thermal equation for the armor surface
area.
For radiators in which (dQ)/(dQc) is constant, the surface exposed
by the armor to space in the axial distance from the radiato~~k%et
to the station at x is given by
iC
a,x = P 1 1 T
S a,x + Te Ta,en - Te (dQ>/(dQ$ DE 4T,3 T a,en + Te Ta,x - Te
T + 2tan'1 a9x ( 1 - 2tan-1
T a,en
( II +
Te Te
The total surface of the armor, S,, is obtained by substituting
T a,ex for T, x wherever T, x appears in eauation (34). 9 9 L On performing
the substitution of T,,,, for T, x 9 in (341, and on then re-arranging the
resulting expression, the following equation is obtained for (dQ)/(dQE) :
30
1
WC(dQ)/(dQ;)]
. 1 + mcp-
1 T a,ex + Te Ta,en - Te
s, U-E 4Te3 T a,en + Te Ta,ex - Te
+ 2tan'1 T a,ex
( )- 2tan'1
Te
7
+ * (35
/
Equations (32), (33) and (35) are three simultaneous equations
involving the unknown, (dQ)/(dQi). It is recalled that in these
equations, mc p is an input quantity and the numerical values of
U and S, are known from previous calculations. The solution for
(dQ)/(dQ;) 1 s obtained from equations (32), (33) and (35) by
iterating with trial values of (dQ)/(dQg) , as follows:
A trial value of (dQ)/(dQg) is assigned and the corresponding value
of U/[(dQ)/(dQE)] is computed. The value of U/[(dQ)/(dQc)] is
inserted into equations (32) and (33) and these equations are solved
for T,,en ad Ta,ex l The values of U/[(dQ)/(dQE)] , Ta,en
and Ta,ex are then inserted into the right side of equation (35),
and the trial value of (dQ)/(dQ;[) is inserted into the left side of
31
equation (35). The numerical values of the left and right sides
of equation (35) are then compared. The value of (dQ)/(dQE) that
makes the left and right sides of equation (35) numerically equal to each other, within the desired degree of accuracy, is the
solution for (dQ)/(dQt).
When the solution for (dQ)/(dQg) has been determined, the associated
value of U/[(dQ)/(dQg)] is inserted into equation (30). A series of
values is assigned to Tg,x in the range Tg,en 2 Tg x 2 T 9 g,ex' and for each assigned value of Tg,x the associated value of Ta,x
is computed with equation (30). In this way a series of paired
values (Tg,x, T,,,) is obtained.
The location, x, at which each combination (Tg,x, T, x) OCCUTS is 9
obtained by inserting the value of T, x 9 into equation (34), computing
S a,xy and solving for x with the relation
X S m = a9
'a (36)
Z
The foregoing procedure yields a unique solution for (dQ)/(dQc), and
numerical values of armor and gas temperatures at known positions
along the tube length.
With the axial distribution of gas and armor temperatures known, it
is possible (by joint use of these temperature distributions, the
perfect gas law and the pressure drop equation in differential form),
to check and refine equations Cl), (2) and (3), and thereby to
produce more exact solutions for N, S,, Ea, (dQ)/(dQg), Tg,x , and
T a9 . When the pressure drop of the gas is small, it is adequate
32
to refine only Tav and Tfilm , equations .(2) and (3).
The radiator sizes and weights presented in this report correspond
to the initially assumed value of Pav/(Tav/Tfilm) , computed on the
basis of equations (1) - (3). Check calculations showed that the
radiators of this report are about 3 to 5 percent larger and
heavier than would be computed on the basis of a more exact value
Of P av/(TavA’film) .
With (dQ)/(dQz) and armor temperatures determined as in the foregoing
section, the calculation of the external fins is performed by joint
use of (dQ)/(dQg) , Ta,x , and independently available data that
relate the fin dimensions to their thermal performance as measured by
(dQ)/(dQ;). The calculation procedures employed for central and
circumferential types of external fins are indicated in the following
sub-sections:
a)Centralfins: A map of LF/Ra versus (dQ)/(dQg) , with Nc,LF
as parameter, is shown for central fins in figure 7. The lines in
figure 7 are based on the numerical values reported in Reference 2.
For converting the values of Reference 2 to the form shown in
figure 7, the following formula was applied to the data presented
in figures 2, 3 and 4 of Reference 2:
(dQ) (dQif)
33
in fig. +
2 1 t-
P Ra'LF
in fig. 3
This formula was obtained by dividing both sides of equation (24b)
of Reference 2 by the quantity vR,/2LF . Figure 7 shows that
LF/ Ra varies linearly with (dQ)/(dQ{) when Nc,LF is held constant.
The values in figure 7, taken as they are from Reference 2, auto-
matically include the effects of temperature drop along the transverse
dimension of the fin, and the effects of radiation interchange
between fin and tube surfaces.
To determine the fin length LF , the abscissa scale of figure 7 is
entered at the known value of (dQ)/(dQg), and for any chosen Nc,LF the
value of LF/Ra is read from the ordinate scaie. The fin length is
given by
LF = (+)Ra = (-)4 (37)
In the present study, N, L 'F
and LF were kept constant for each
radiator. The quantity Nc,LF was varied parametrically.
34
For known LF, the thickness of the external fin at station x is
computed from the definition of Nc,LF,
which yields
N 2a Ta3, LF2
= 9 c,LF
kFnF,x
'F,x = 20.Ta3, LF2
9
kF 'c L 9 F
(38)
(39)
Equation (39) shows that when LF and N, L are both kept constant, 'F
the fin thickness changes with the armor surface temperature. Since
T a,x decreases axially, the fin thickness decreases from entrance to
exit of the radiator. This is true only for radiators like those of
the present study, in which (dQ)/(dQE) , LF , and Nc,LF all have
values that remain the same from one axial station to the next
along the armor surface.
The transverse span of a single centrally finned tube is (Da + 2LF).
For N tubes in parallel, the total span is N(D, + 2LF), which is the
header length for centrally finned radiators.
b) Cirm1 exteraaLfiaS : A typical map that relates
the spacing of circumferential external fins to (dQ)/CdQt) and to N, R 9 a at a fixed value ofRo/Ra is presented in figure 8. The data in
figure 8 are based on the results reported in Reference 3 and are
35
basically identical with the data of that Reference. In slight
variations from the form employed by Reference 3 to present the
data, different nomenclature is used herein, and figure 8 employs
the ratio of fin spacing to fin inner radius as the curve identifi-
cation parameter, instead of the ratio of fin spacing to fin outer
radius employed by Reference 3. As the fin inner radius equals
Da/2 and is known explicitly from previous calculations, the inner
radius is convenient for the present calculations and for this
reason is used in the denominator of the curve identification
parameter in figure 8.
The fin outer radius, fin axial spacing, fin thickness and number
of fins per tube are determined as follows:
The ordinate scale in figure 8 is entered at the known value of
(dQ)/(dQ;) , and a line is drawn parallel to the axis of abscissas.
This line intersects one or more curves of the figure, and each
intersection point determines a combination of numerical values,
SF/R, and Nc,R, l It is evident that when (dQ)/(dQc) is held constant,
the consideration of more than one value of sF/'Ra is equivalent to
parametric variation of Nc,R; With sF/Ra and NC R both known, a 9 a
the fin dimensions and spacing are computed with the following
formulas:
The fin outer radius is given by
R, = ()Ra = w+- (40)
36
The fin axial spacing is given by
SF = (41)
The definition of the conductance parameter Nc,R for circumferential a
external fins is
(42)
As indicated above, the numerical value of N, R is known for each )a
combination of R,/R, , (dQ)/(dQE> and sF/R a'
The fin thickness at station x is computed by the formula
AF,x = 2oT,:x Ra2
kF Nc,Ra (43)
In the present study, N, R is kept axially constant in each 3 a
radiator. Since Ta,x decreases along the armor surface, the fin
thickness AF,x decreases steadily from entrance to exit stations
along a radiator tube. This is true only for radiators like those of
the present study, in which (dQ)/(dQE) , R,/R, , SF/R, and N, R )a
are constants for the entire radiator.
37
The number of fins per tube is given by the adequate approximation
Z Y =
SF + 'F,av (44)
in which AF,av is the arithmetic average of the fin thicknesses
at the entrance and exit stations of the radiator tube.
Note is taken that the foregoing formulas have been illustrated with
curves for a single value of R,/R, (figure 8). In an exhaustive
optimization study of circumferential external fins, exploration of
several values of R,/R, is required, in search for the optimum value
of R,/R,.
The transverse span of a single finned tube is 2R,. If N tubes are
arranged in parallel in one plane so that the fins of adjacent tubes
just touch each other, the combined transverse span of all the tubes
is N(2Ro), and this is the minimum possible header length of a
circumferentially finned radiator in which the external fins do not
mesh with or overlap each other.
38
Headers
In this study,the headers are designed for axially uniform drain-off from
the supply header and axially uniform feed into the exhaust header.
With the origin of x taken at the entrance plane of the supply
header or, equivalently, at the exit plane of the exhaust header
(figure 21, the conditions for uniform drain and uniform feed are
expressed by the equation
dm .
- = - mH,en =- tiH7ex = const . dx LH LH
(45)
In addition, the condition is imposed that in each header
the mass flow per unit cross-sectional area shall have the same
value at every axial station of that header. This condition is
expressed by the equations
Gx = mH,en = G en = const.
A en (46a)
. mX = Gen Ax
39
G, = mH,ex
A = G,, = COnSt.
ex
. mX = Gex Ax
Solution of equations (49 and (46) yields
dH,x =
X dH,x =
d H,ex 1 - -
LH
(46b>
(47a)
(‘+7b)
In eauations (45) and (47) the header length LH is known, as was
indicated at the ends of the sub-sections on central and circumfer-
ential external fins; formulas for LH are itemized explicitly soon
hereafter. The diameters dH,en and dH ex are initially unknown; they , are computed by solution of the pressure drop equations for the gas
in the supply and exhaust headers. Heat transfer in the headers
is ignored in the present study, and the effect of heat transfer on
the header diameters is not coneidered.
The gas flow in each header is treated separately and as though
the flow were incompressible; different gas densities, based on
the respective gas temperature-and-pressure combinations, are
employed for the two headers. In each header the gas filament
40
that flows the full length of the header is assumed to experience
a pressure drop based on three factors: (1) friction, (2) a loss
of one dynamic head based on the velocity in the header, and (3) a
loss of one dynamic head based on the velocity in the radiator tube
and postulated to occur during passage from the supply header into
the radiator tube or from the tube into the exhaust header. Gas
filaments that flow only a portion of the length of the header are
assumed to have the same pressure drop as the filament that flows
the full length of the header; the smaller friction pressure drop
in the flow along only a portion of the header length is assumed
to be supplemented by pressure drop in calibrated orifices at the
entrances and/or exits of the tubes. The friction component in the
length interval (x, dx> is computed herein with the formulas for
turbulent flow:
dx dPfr = -4f (3X2
dH,x 2w
0.046 f =
(Gx d~,~/ct) Oe2
(48)
For each header P is treated as a constant; p is taken equal to
P en in the supply header, and is taken equal to pex in the exhaust
header.
In order to compute the friction pressure drop of the gas filament
that flows the full length of the header, equation (48) is integrated
from x = 0 to LH9 making use of equations (46) and (47). The complete
41
pressure drop of the gas filament is then obtained by summing the
friction term and the pertinent dynamic head losses, which are
given by the following expressions:
The dynamic heads in the respective headers are given by
G2 'Hfen 1 =
2g 'en 2g Pen CTi412 d$,en
G2 'Hfex 1 =
2g pex 2g Pex(n/4)2 di ex ,
(49a)
(‘+9b)
The dynamic heads based on the velocities in the radiator tube at the
entrance and exit stations of the tube are given by
2
Tube entrance dynamic head = Gtube 2g Pen
2
Tube exit dynamic head = Gtube
2g Pex
(50a)
(50b)
By setting the sum of the friction pressure drop and dynamic head
losses in each header equal to the allowable pressure drop in the
header, the following equations are obtained:
42
For
LH d4.8
H,en
+
2
+ Gtube -- 2g Pen
(Allowable supply header Ap)
and for the exhaust header,
0.2 -1.8 -
2.5 4( l 046) wex mH,ex
2g PexW4P -
2 - + mH,ex
Pex(-rr/4)2
1 LH +
4.8 dH,ex
1 1 2
+ Gtube 4
dJi, ex 'g Pex
= (Allowable exhaust header Ap>
( 51a>
(51b)
Equations (51a) and (51b) permit solution for dH en and s,ex, ,
respectively. In these equations, the allowable pressure drops
for the supply and exhaust headers, the gas densities pen and p,,,
and the viscosities pen and IJ-,, in those headers, are known as
input. The quantity Gtube is known from previous solution of
43
equation (21). The quantities kH,en and rkH,,x are the total gas
flow rates per branch in the entrance and exit headers. For
unsplit headers as in figure 2a,
. mH,en = 'H,ex
(52a) = ti of entire radiator
.For split headers as in figure 2b,
'H,en = 'H,ex
= 2-(. m of entire radiator) (52b) 2
If the headers were split into 2n branches, the flow rates per
branch would be given by the relation
. mH,en = tiH,ex = ti of entire radiator)
In equations (51a) and (sib), the header length LH is as follows:
For central-tee external fins (figure 4),
LH = N(D, + 2LF) (Unsplit headers) (53al
LH = #(Da + 2LF) (Once-split headers) (53b)
44
For circumferential-type external fins (figure T), the header
length depends on the tube spacing required to avoid excessive
mutual occlusion of the finned tubes when arranged in parallel.
The minimum possible tube spacing for non-meshing fins is such
that the fins of adjacent tubes just touch each other. Thus,
for circumferentially finned tubes,
LH > N(2Ro) (Unsplit headers) (%a)
LH 1 (Once-split headers) (fib)
In the present report the header lengths for circumferentially
finned tubes were taken at the values corresponding to tangency
of the fins of adjacent tubes; that is, the "equal" signs were
used in equation (54). The extent to which the thermal perfor-
mance of an array of N closely spaced tubes differs from the
summed thermal performances of N isolated tubes requires detailed
analysis outside the scope of the present report.
It may be noted that for headers split into 2n branches, the
denominators in the right members of equations (53b) and (5&b)
would be 2n instead of the value 2 now shown.
With d and d H,en H,ex
known by solution of equations (51a) and
(Qb), the total surface exposed by both headers combined is
computed with the following formula, obtained by integrating
45
elements of surface "(dH,x + 26,)dx aS x goes from 0 to LH,
k 2 SH = vLH,unsplit 3 - dH,en + dH,ex + 2
In which LH,unsplit is given by equation (53a) or (54a).
Equation (55) applies both to unsplit headers and to
headers split into any even number of identical branches, since
the product 2n(LH ,unsplit/2n), which ari ses during consideration
of split headers, always reduces to LH,unsplit. The effect of
splitting the headers is reflected in the diameters dH,en and dH,ex 3
which become smaller as the number of header branches increases.
In equation (55) the armor thickness 6, is taken equal to the
value earlier obtained from joint solution of equations (23) and
(27). A refinement of 6a is considered in the following section.
Weights and Planform Area
The radiator total weight is the sum of the component weights, which
are determined as follows:
The total surface exposed by the armored tubes and headers is given
by the sum of the individual surfaces,
'a,total = sa + SH (56)
46
in which S, is given by equation (27) and SH by equation (55).
Equations (27) and (55) are both initially based on the unrefined
armor thickness 6, obtained by joint solution of equations (23)
and (27). A refined value of 6, is obtained by inserting Sa,total
into equation (23):
'a,total = 'acSa,total) 113 B
(57)
with Ca given by equation (24).
A more highly refined value of the armor thickness is obtain-
able by inserting the 6altotal of equation (57) into the formulas
for S, and SH , thereby refining Sa,total and, through (57), 6a,total'
With a more accurate value of 6a total thus available, the tube and , header weights are as follows:
= Pa[NZr(di + 6a, total ) 6a,total] (58)
weight Tota1 header) = Pap6a,total LH,unsplit ] x
I
(59) x
'a,total
In equation (591, the bracketed volume term is obtained by integrating
volume elements of the form T(dH x + 6a totalj6a totaldX as x t J , goes
I -.-
47
from 0 to LH in each header. Equation (59) applies both to unsplit
and split headers, for the same reason as was given in connection
with equation (55).
The weight of the internal fins is given by
( 1 PfNZ
<urn of the cross-sectional areas of all fin metal parts, including fin bases, com- puted at any single tube cross section taken perpendicular to the axis of the tube. -
(60)
If the internal fins are brazed to the tube walls, the weight computed
with equation (60) may be multiplied by 1.1 in order to make approx-
imate allowance for the weight of the braze metal. If a flow block-
age tube is present in the interior of the radiator tube (figure 3c),
the weight of the blockage tube must also be included. In this
study, no allowance was made for braze metal weight, but when a flow
blockage tube was assumed to be present the weight of that tube was
taken into account. The thickness of the blockage tube wall was taken
as .005 inch and its material was assumed to be aluminum of density
172 lb/f&
The weight of the external fins is affected by the fact that in the
class of radiators studied, the thickness of the external fins
decreases axially from entrance to exit of the radiator. Thus in
computing the weight a properly averaged fin thickness must be
48
employed. When an average fin thickness is used, the weight of
central-type external fins is given by
Central ( 1 fins (61)
The average thickness of axially tapered central-type external fins
is given by the formula
1 1
'F,av = 2 'F,xdx Central ( 1 fins (62)
in which AF,x is given by equation (39). For the present study,
it was convenient to use an approximate value of AF,av , rather than
the one defined by equation (62). The following approximate formula
was employed:
&
'F,av - cAFjat x=~+ 1*75(AF)at x= Z
2.75 (63)
Check calculations were made for the purpose of comparing the values
of AF av given by equations (62) and (63). The calculations showed
that eiuation (63) yielded values of A ~,a~ withinf5$ of those computed
49
with equation (62), and that the associated overall weight uncer-
tainty was less than fl percent of the total radiator weight in
the cases of interest. Hence for the exploratory purposes of the
present study, the use of equation (63) for AtF av was thought to 9
be acceptable and equation (63) was employed herein.
In the case of circumferential external fins the total fin weight
Figures 17 and 18 show that for the 25 it long internally finned
radiators, the effect of Nc,LF on total radiator weight is small
in the range of Nc,LF considered; this was also true for the 25 ft
long internally bare radiators of figure 11. Careful comparison
of figures 17 and 18 with figure 11 discloses, however, that the
Nc,LF for minimum weight of the 25 ft long internally finned tubes
is substantially higher than the 0.3 weight-minimum value of Nc,LF
for the internally bare radiators of figure 11. The change in welght-
optimum N,,LF Is due to the substantially larger diameters of the
Internally finned tubes, as shown for the 25 ft long radiators In
Table 3, below. Increases In tube diameter, with attendant decreases In
the number of tubes and In the length and weight of the headers, cause
the headers to become lighter than the external fins (or alternately,
cause the external fins to become heavier than the headers). Since
Increases In Nc,LF produce decreases In external fin weight, the
weight-optimum value of N c,LF moves toward higher values as the
tube diameter Increases. Thus, at fixed tube length, the effect of
Internal fins Is to Increase the weight-optimum value of Nc,LF ,
In comparison with the best value for Internally bare tubes.
Table Tube Lengths-and Diameters In Several --Weight Ram
No. of internal Tube i.d. Tube o.d. 1 Inch)
11 25 None 1.07 1.48
17 25 10 2.05 2.44
18 25 8 2.98 3.39
19 6 30 1.23 1.63
20 6 12 1.29 1.68
1. 20 1.48 1.87
IC-
70
On the other hand, figures 19 and 20 show that if the weight-optimum
tube diameter of internally finned radiators is reduced by reducing
the tube length to 6 ft (Table 31, the header weight dominates over
the external fin weight. Thus, to achieve minimum radiator weight,
NC&F must decrease to values lower than those that are optimum
for Internally finned radiators with 25 ft long tubes. In figures
19 and 20, the weight-optimum values of N, LF are all close to 9 the 0.30 value which was previously found to be optimum for
25 ft long internally bare radiators.
The foregoing discussion indicates that Internal finning of the
radiator tubes results in a marked tendency of the weight-optimum
N C&F
to increase. The discussion also shows, however, that if
the optimum diameters of internally finned tubes are reduced by
means of reductions in the tube length, the weight-optimum N c&F
can be maintained at a low value, with substantial attendant benefits
in radiator weight and planform area.
In figures 17 - 20, explicit note may be taken that reductions in
N c&F ' reduce decreases in the weights of the headers. Reductions
in N c,~F , together with optimization of the number of header
branches as discussed earlier in the text, are two effective means
for reducing header weight.
71
Existence of optimum number of internal fins per tube: For internally
finned radiator tubes it is desirable to Inquire whether there exists
an optimum number of fins per tube. This question may be discussed
by use of figures 15, I6 and 20. Figure 15 corresponds to 6 ft long
tubes with radially long Internal fins (figure 3c), and figures I6
and 20 correspond to 6 ft long tubes with radially long-and-axially
interrupted Internal fins (figure 3d).
Figure 15 shows that at each of two different numbers of Internal
fins per tube (20 and 301, a weight-optimum tube i.d. occurs, and
that the optimum 1.d. of the tubes with 30 internal fins Is notlce-
ably larger than the optimum l.d. of the tubes with 20 Internal fins.
(The growth In weight-optimum tube diameter with Increases in the
amount of internal fin surface has been discussed previously In the
text.) In addition, figure 15 shows that the minimum-weight radiator
with 30 fins per tube Is lighter, and has a significantly smaller
planform area, than the minimum-weight radiator with 20 fins per tube.
Thus figure 15 shows that one number of Internal fins can be better
than another from the viewpoints of both radiator weight and size.
72
On the other hand, figure 16 shows that for an internal fin
geometry which differs from that of figure 15, three different
Internal fin numbers per tube (12, 16 and 201, yield respective
minimum radiator weights that are Indistinguishable from one
another. The planform areas, however, are not all the same; the
planform area decreases as the number of Internal fins per tube
increases. Thus, one number of internal fins may be better than
another number from the viewpoint of radiator size. It can be
shown as follows, however, that even In a range where significant
changes In the internal fin surface per tube appear to produce no
effect on radiator weight, there does exist a weight-optimum
number of Internal fins per tuber
It is noted that for all the radiators of figure 16, N c&F is equal to 1.0 . The discussion in the foregoing sub-section has
shown that the weight-optimum Nc,~F changes as the tube diameter
changes. This indicates that Nc,~F can be used to Identify the
weight-optimum number of internal fins per tube, as illustrated
by figure 20. Figure 20 shows that when optimized with respect
to Nc,~F as well as with respect to tube i.d. (figure 161, the
minimum-weight radiator with 12 internal fins per tube is lighter
than the minimum-weight radiator with 20 fins per tube. The
radiator with the larger number of Internal fins per tube remains
the smaller in planform area, however, at all values of Nc,~F
In the range shown in figure 20.
73
The foregoing discussion indicates that for each type of internal
fin geometry there exists a weight-optimum number of internal fins
per tube. At each prescribed set of radiator operating conditions
and tube length, the optimum fin number per tube may be identified
by parametric exploration of that fin number, and by optimization
of both the tube i.d. and NC L
fins per tube. 'F
at each assigned value of internal
Minimum-weight internally finned radiators: For internally finned
radiators with central-type external fins, the minimum specific
weights and associated specific planform areas computed in the
present study were as follows:
Table 4. Minimum Weight Internally Finned Radiators With Central External Fins
Tube Internal No. of fins Radiator Sp. planform area - length fin type per tube sp. wt.
(ft) (lb/me) (inT;;2h;dTrs)
e
25 Long radial (figure 3c)
10 83.3 45.7
6 Long radial (figure 3c>
30 82.1 33.7
6 Interrupted (figure jd)
12 78.4 33.0
The headers of the radiators in the foregoing table are split once.
In the case of the radiator of 25 ft tube length, the combined
weight of the supply and exhaust headers is 73 lb, hence only small
74
gains could be achieved by further header subdivision. In the
case of each radiator of 6 ft length, however, the header weight
is about 225 lb. For the radiators of 6 ft tube length, sub-
division of each header into 4 branches (instead of the 2 branches
that underlie the above-tabulated radiator weights) would reduce
the header weights by about 70 lb, and the radiator specific weights
by about 7 lb/me.
The planform areas in the foregoing table include the incremental
projected areas contributed by the headers, conservatively based
on the largest occurring header diameters. In the case of the
radiator with 25 ft long tubes, the header area contribution is
only about 3 percent of the basic panel area of the tubes plus
their external fins. In the case of the radiators with 6 ft long
tubes, however, the header area contribution is about 13 percent
of the tube-fin panel area. Inasmuch as the headers and tube-fin
panel must be housed in the same vehicle, it appears reasonable
to include the header area in the total (projected) planform area.
The radiator specific weights and sizes in the foregoing table may
be compared with the 95.1 lb&W, and 53.3 ft2/KWe of the optimum
internally unfinned radiator of the present study. Such comparison
shows that for the class of radiators and operating conditions
considered, internal finning of the tubes results in weight reductions
of 12 to ia percent, and size reductions of 14 to 38 percent, in
comparison with the weight and size of the optimum radiator with
75
internally unfinned tubes. If the headers of the lighest internally
finned radiator in the foregoing table were split twice instead of
once, that radiator would be about 25 percent lighter and about
40 percent smaller than the optimum internally unfinned radiator of
the present study.
Radiators With Circumferentz. External Fins
The parts of the external fins that are distant from the surface
of the tubes depend upon conduction for most of the heat that
reaches them. Hence there is a substantial temperature decrease
along the fins in the direction away from the tube surface. In
the case of central-type external fins, the average temperature of
the body of the fin in the neighborhood of the supply header is
lower than the temperature of the header. In that region of the
radiator, the fins and the fin-to-tube junctions are subject to
tensile stress. Similar stress may exist in the neighborhood of the
exhaust header. In a radiator with non-isothermal working fluid,
there is also a temperature decrease in all metal parts in the
direction of fluid flow. In the case of central-type fins that are
continuous, constrained plate-type deformation or stress may arise
from the simultaneously occurring axial and transverse temperature
fields. Thus, radiators with central-type external fins may
operate with substantial stress at the fin-to-tube junctions or
in the body of the fins. These stresses are probably of greater
significance than those that exist in the tube walls as a result of
circumferential temperature non-uniformity in centrally-finned tubes.
76
In the case of circumferential, i.e., annulus shaped, external
fins (figure 51, the radial temperature drop in the fins that
arises from outward conduction of heat leads to a compression of
the fins at their junctions with the tubes. Under compressive
force, separation between fin and tube appears unlikely. In
addition, when each fin is a separate unit, temperature differences
between fin and header, or between one fin and another, do not
give rise to fin-to-tube junction stress or to added stress within
the body of the fin. Further, with a circumferential arrangement
of the fins, the temperature in the tube wall tends to be uniform
around the circumference. Thus, radiators with circumferential
external fins may be significantly less vulnerable to thermal
stress than are radiators with central-type external fins.
Circumferentially finned tubes also appear to offer relative ease
of fabrication; and the fins themselves may perform a non-negligible
bumper function against obliquely arriving meteoroids and thereby
may permit reduction in the thickness of the armor.
Accordingly, it is of interest to make exploratory calculations of
the sizes and weights of tubular radiators equipped with circum-
ferential external fins. Results of preliminary calculations for
such radiators are presented in figure 21.
Figure 21 presents calculated weights and planform areas for two
sets of radiators equipped with circumferential fins of radius ratio
RoDa = 4, with the fins of adjacent tubes just touching each other.
One set of curves in figure 21 corresponds to the internally
unfinned radiator of 25 ft tube length and 1.07 inch tube 1.d. that
77
was previously optimized in figures 10 and 11. The second set of
curves corresponds to the internally finned radiator of 6 it tube
length and 1.29 inch tube 1.d. with 12 axially interrupted internal
fins per tube that was previously optimized in figures 16 and 20.
The two sets of radiators in figure 21 have the following properties:
(a) For each set of radiators, the number of tubes, the tube length
and the tube inside and outside diameters are the same for the
circumferentially finned geometry as they were for the centrally
finned geometry. Hence, for each set of radiators the armored
tube weight is constant and equal to the tube weight in the centrally
finned geometry.
(b) For each set of radiators, the header lengths and weights
and the radiator planform area are constant. This follows from the
fact that the number of tubes, the tube outer diameter, the ratio
Ro/Ra and the lateral spacing between tubes are all constant.
(c) The overall convection-conduction coefficient U , the armor
temperatures, and hence also the ratio (dQ>/(dQE), are the same for
the circumferentially finned radiators as for the centrally finned
radiators. That is, each set of radiators has a fixed combination of
values of Da (or Ra), R,/R, , and (dQ)/(dQE) . Therefore, as was
discussed in the Calculation Procedure , parametric variation of NC R 9a is equivalent to parametric variation of the fin axial spacing (sy).
In addition, as NC R 9 a
is varied, the fin thickness (A, ) varies in
accordance with equation (43) of the Calculation Procedure. Hence,
parametric variation of N,,R, implies definite variations in the
spacing, thickness and axial pitch (SF + AR) of the external fins,
under the conditions governing figure 21.
78
It follows from the foregoing that NC R , a is the only independent
variable in figure 21, and that the radiator weight variations in
the figure are due entirely to variations in the weight of the
external fins.
Figure 21 shows that as Nc,R, increases from an initially low value,
the axial pitch between fins decreases steadily. Since the tube
length is fixed, a decrease in fin axial pitch implies an increase
in the number of fins per tube. With increases in N,,Ra , however,
the fin thickness (not shown separately in figure 21) decreases
steadily. The balance between the increase in the number of fins
and decrease in their thickness leads to a minimum in the weight of
the fins at an intermediate value of N,,R, ; in figure 21, the
value of N,,R, at which the minimum fin weight occurs is .03 for
both sets of radiators shown in the figure. Since the fins are the
only component that can affect the total radiator weight under the
conditions of figure 21, the radiator weights also have their
minimum values at NC R 9 a = .03 . The following table presents data
of interest for the minimum weight radiators of figure 21.
le 5 a Weight Radiators With Swerential External Fu
Tube Internal Internal Radiator Sp. planform area length fin type fins per tube sp. wt. (incl. headers)
(ft) ( lb/me) (ft2/KWe>
6 Interrupted 12 81.6 23.1 (figure 3d)
25 None None 102.5 36.8
79
The values in Table 5 make no allowance for mutual shadowing of
the finned tubes, nor for possible weight reductions due to the
bumper effect of the circumferential external fins. The values
in Table 5 are, therefore, only tentative. Thus, tentatively, the
table shows that the internally finned radiator of 6 ft tube length
is about 20 percent lighter and more than 35 percent smaller than
the optimum internally unfinned radiator of 25 ft tube length. These
percentages are substantially the same as the corresponding ones for
centrally finned radiators.
A comparison between the minimum-weight internally finned radiators
of 6 ft tube length in Tables 4 and 5 indicates, tentatively, that
the radiator with circumferential external fins is about 5 percent
heavier and about 30 percent smaller than the radiator with central-
type external fins.
Comparison of weights and sizes also indicates, tentatively, that
the minimum weight internally finned radiator of 6 ft tube length
with circumferential external fins is about 15 percent lighter and
about 55 percent smaller than the optimum internally unfinned
radiator of 25 ft tube length equipped with central-type external
fins.
80
CONCLUDING REMARKS
A preliminary study has been made of Brayton cycle radiators that
use a gas as their working fluid. The radiators have been assumed
to be assemblies of armored, externally finned tubes that lie in
one plane and radiate heat to both sides of the plane. The radiator
operating conditions that have been assumed correspond to a solar-
powered Brayton cycle that uses argon as working fluid and delivers
10 kilowatts of electrical power steadily during a 365 day mission,
in an environment in which protection against meteoroids is a
substantial requirement.
One purpose of the study was to develop a method of calculating the
sizes and weights of radiators of the sort described in the preceding
paragraph. A method of calculating such radiators has been presented.
Another purpose of the study was to determine whether significant
effects on radiator size and weight result from the use of finned
heat transfer surface inside the radiator tubes. For this purpose,
four internal fin geometries have been evaluated in radiators
equipped with conventional central-type external fins.
A third purpose of the study was to consider briefly the effects
on radiator size, weight and stress that might result from the use
of circumferential (annular) radiating fins on the external surfaces
of the radiator tubes. Illustrative results for radiators equipped
with circumferential external fins have been presented.
81
The principal findings of the study are as follows:
(a) The main effect of internal fins is to reduce substantially the
radiator planforn area; to a lesser but non-negligible extent,
internal fins also reduce the radiator weight. .The numerical results
Indicate that optimized radiators with internal fins can be more than
35 percent smaller in size and more than 15 percent lighter in weight
than optimized radiators without internal fins.
(b) Circumferential external fins may offer relative ease of fabric-
ation, relative freedom from thermal stress, and a bumper effect
against obliquely approaching meteoroids. If tube-to-tube occlusion
does not necessitate wide spacing between tubes, circumferential
external fins may offer worthwhile reductions in radiator size. With
occlusion neglected, a 30 percent reduction in planform area was
computed on substituting circumferential for central-type external
fins in the smallest (internally finned) radiator studied.
As part of the study leading to the foregoing findings, the following
were done:
The numerical results were employed to demonstrate that there exist
optimum values for the independent geometric variables of the
radiator, namely, tube length, tube inside diameter, number of
internal fins per tube for each species of internal fin geometry,
and number of branches into which the headers are split. It was also
indicated that an optimum value exists for the conductance parameter
of the external fins, and that the optimum value is affected by the
diameter of the tube.
82
The effects of internal fins on the weight-optimum values of.the
independent geometric variables were discussed. It was indicated that
(1) The weight-optimum length of internally finned tubes is shorter
than that of internally unfinned tubes .
(2) At fixed tube length, the weight-optimum diameter of internally
finned tubes is larger than that of internally bare tubes; but if
a relatively short length is used for internally finned tubes,
then the weight-optimum diameter is about the same as that of
internally bare tubes.
(3) If a relatively short tube length and associated optimum inside
diameter of internally finned tubes are both used, the optimum
value of the conductance parameter for the external fins is
about the same for both internally finned and internally bare tubes.
(4) The optimum number of header subdivisions is significantly larger
for internally finned radiators than for internally bare radiators.
83
APPENDIX A
A
a
b
'a
CT
C
cP D
d
dQ
dQb
dQ;
h
flow area, ft2
correction factor for finite plate thickness and for spalling, 1.75, non-dimensional
radial length of internal fin, ft
coefficient in armor thickness equation, fto l 502
temperature coefficient, non-dimensional
speed of sound in armor material, ft/sec
specific heat at constant pressure, Btu/lb, OR
outside diameter, ft
inside diameter, ft
heat radiated by an infinitesimal surface element, consisting of armor surface-plus-external fin surface, in an externally finned radiator, Btu/hr
heat radiated by an infinitesimal element of bare armor surface In an externally unfinned radiator, Btu/hr
heat radiated by an infinitesimal element of bare armor that has the same surface area and the same surface temperature as the armor of the externally finned element which radiates heat dQ defined above. The numerical value of dQg is equal to
0~ (‘Jayx - Te4) dsa 3 in which E , T,,, , T, and dS, are the same as those of the armor surface in the externally finned element which radiates the heat dQ defined above, Rtu/hr
modulus of elasticity, lbf/ft2
friction factor, non-dimensional
mass flow rate per unit flow area, lb,/hr ft 2
gravitational conversion factor, 32.2 x (36C~o)~ , (lbm/lbf)(ft/hr2>; in eq. (251, g = 32.2 (lbm/lbf)(ft/sec2>
convective heat transfer coefficient of gas, Btu/hr ft2 oR
I --
84
k thermal conductivity, Btu/hr ft2 OR ft'l
L length, ft
1 length of radiator tube, ft
i
N
NC
mass flow rate, ib,/hr
number of radiator tubes, non-dimensional
black body conduction parameter of external fin, non-dimensional
N 'hLF NC based on fin length, 20 Ta3 LF2/kFAF, non-dimensional
N c9Ra NC based on armor radius (inside radius of circumferential fin), 20 Ta3 Ra2/kF AP, non-dimensional
n number of internal fins per tube; also, half the number of branches of a header split into 2n branches, non- dimensional
Pr Prandtl number, non-dimensional
pW
wetted perimeter, ft
P(0) zero penetration probability, non-dimensional
P static pressure, lbf/ft2
Q total heat release; heat release of externally finned radiator, Btu/hr
9 heat released by a single tube, Btu/hr
R radius, ft
r tube inside radius, ft; also, thermal resistance, 'R/(Btu/hr)
Re
S
S
T
Reynolds number, non-dimensional
surface area, ft2
axial spacing of circumferential external fins, ft
temperature, OR
u heat transfer coefficient referred to outer surface of armor, Btu/hr ft2 oR
ii mean speed of meteoroids, ft/sec
V gas speed, ft/hr
X distance from radiator tube entrance plane; or from entrance plane of supply header, ft
Greek symbols:
meteoroid mass distribution constant, 5.3 x lo-llgmS/ft*-day
meteoroid mass distribution constant, 1.34, non-dimensional
thickness of external fin, ft; drop (In pressure)
thickness of armor or internal fin, ft
emlsslvlty, non-dimensional
efficiency, effectiveness, non-dimensional
exponent on the speed ratio v/c, non-dimensional
dynamic viscosity, lb,/hr ft
number of circumferential external fins per tube, non- dimensional
mass density, lbm/ft3
Stefan-Boltzman constant, 0.171 x 10s8 Btu/hr ft* OR4
time, days
Subscripts:
a armor
av average
b bare armor radiator
e environment
eff effective
en entrance station
eo. equivalent
ex exit station
F external fin
86.
f internal fin
film fluid film
flow flow area
fr friction
g gas
H header
i inside; inner surface of tube
LF based on length of external fin
momentum arising from change in fluid momentum
0 outside
P particle (meteoroid)
Ra based on armor radius
split relating to headers split into two or more branches
t target
total based on combined contributions of armored tubes and headers
tube pertaining to radiator tube
unsplit relating to unsplit header
W wall; wetted
X at a station distant x from radiator tube entrance plane, or from supply header entrance plane; also, %p to station x" when applied to armor surface (Sa,x)
87
APPENDIX B
DERIVATION OF ARMOR THERMAL RELATIONS
The relations presented in the text for U, T, x and Sa x, equations
(281, (30) and (341, respectively, are derive: in the iresent
appendix. For this purpose a representative tube like the one
shown in figure 22 Is considered. Figure 22 shows tube internal
details representative of those considered in this study, but gives
no details of external fins that would normally be present on the
outer surface of the armor. The external fins are taken into
account on a generalized basis by the parameter (dQ)/(dQE). This
parameter encompasses a large variety of external fin geometries
without need for detailed specification of those geometries.
Derivation of the expression for U: Heat balances, expressed in
terms of component and overall resistances to heat flow, are
employed.
Steady state, one-dimensional heat flow is assumed. The inner and
outer wall temperatures of the tube are assumed to be circumferen-
tially uniform, and the temperature of the gas within the tube is
assumed to be uniform over the cross section of the tube. Heat
flow from the gas to the outer surface of the armor in the length
interval (x, dx) is considered.
The resistance to heat flow from the gas to the inner wall of the
tube is expressible as
T rg,x =
g,x - Ti,x dQ
(Bl)
In equation (Bl), rg,x includes the resistance to heat flow from
the gas to the fins. The quantity Ti,x is the temperature of the
inner wall of the tube proper. For thin fins, the fin bases and
their exposed sides are considered to be part of the inner wall of
the tube and are assumed to be at the same temperature Ti x as the 9 tube inner wall.
The expression for rg,x in terms of the gas heat transfer coefficient
and the effective heat transfer surface is
1 rg,x =
h dSeff
in which dSeff is the element of effective heat transfer surface
swept by the gas. The formula for dSeff is as follows:
dS eff = dSw,i + fl, dSf
(B2)
(B3)
89
The term dSw,i consists of the sum of the exposed poEtion of the
tube inner wall, the exposed portion of the inner surfaces of
the fin bases, and the exposed sides of the fin bases. For fins
as shown in figure 22, the exposed portion of the Inner wall of a
single tube is given by
portion of inner wall, per
) tube
= ($r di) dx
The exposed inner surface of the fin bases, taking account of the
portions covered by the roots of n fins, each of thickness 6f,
is given for a single tube by the expression
Exposed inner surface of bases, per
- 2 6f) -nEf dx 1 The surface exposed by the sides of the fin bases, taking into
account that there is one exposed side per fin, is given by
Surface exposed by sides of fin = bases, per tube
(n ef)dx
The quantity dG,i Is obtained by summing the three foregoing
components of the exposed inner surface per tube, and multiplying
by the number of tubes; thus, d&,i is given by the following
expression:
90
034)
Define
dStube = T di N dx
h q h sf eff S
W,i
With these definitions, the term h dS,ff becomes
(B6)
(B7)
and the equation for the local thermal resistance of the gas becomes
Tg,x - Ti,x = 1 rg,x =
dQ h eff dStube
(B8)
91
The heat flow by one-dimensional conduction across the'wall of a
single armored tube is given by (Ref. 13)
7
dq = 2~ ka (dx) (Ti,x - Ta,x) - I
\ ui I (B9)
ZI 2(~ Da dx)ka cTi,x - Ta,x) -
For N identical tubes the total heat flow -1s
dQ = Ndq = 2(NaD, dx) k, (TI,x - Ta,x)
(BlO)
= 2(dSa)ka (Ti x - Ta,x)
The resistance to heat flow across the armor is then
Ti,x - Ta,x = Da In (D,/di) r a,x = dQ 2k, dS,
(I3111
-_-
92
The total resistance to heat flow from the gas to the outer surface
of the armor is the sum of the resistances of the gas and of the
armor; thus,
r = rg,x + ra,x (B12) X
Combining the expressions in equations (B8) and (Bll), and using
the fact that
dStube
equation (B12) takes the form
rx =
heff (' 6f l-- di
)
1
II
Da P - + dSa di
r (B13)
+ h eff
Also, again by summation of equations (B8) and (Bll),
r = Tg9X - Ta,x X dQ
(I3141
I-- --
93
Equating the right members of (B13) and (B14) and solving for dQ,
dQ = heff (1 - &) (Tg,x - Ta,x) dSa
0315) Da
- + heff -- di
Equation (B15) may be simplified by writing
dQ = u (Tg,x - Ta,x) dSa (I3161
which is a defining equation for U. Comparing equations (B15) and
(B16), the expression for U is
u =
which is the formula for U in equation (28) of the text.
(B17)
Derivation of eauation for armor temperature: The expression for
the local armor temperature as given by equation (30) of the text
is derived in the present sub-section. The derivation makes use of
94
the ratio (dQ)/(dQE) . In this ratio, dQ is the total heat radiated
by an element of armor surface and its external fins, the local
temperature of the armor surface being Ta,x ; the term dQz is the
heat that would be radiated by the same armor surface element if the
external fins were removed and the surface temperature of the armor
were somehow maintained at Ta,x .
By identity,
(de), = (dQ),
(dQ;;) (dQ;lx
X
(B18)
The appearance of the subscript x in equation (Bl8) signifies that
dQ and dQz both change with x. In the general case, (dQ),/(dQ{),
will also change with x.
For a surface element of bare armor operating steadily at temperature
T a,x in an environment of effective temperature T,, the net heat lost
by radiation is given by
(dp;), = CJ~ (T;,x - Tf) ds,
The heat lost by radiation from the externally finned version of the
armor element when operating at surface temperature Ta,x is, by
definition, (dQ), . In steady state, when the heat lost by radiation
is equal to the heat received by convection-and-conduction, (dQ), has
the value given by equation (B16).
95
Substituting equation (B16) for the left member of (B18), and using
equation (B19) in the right member of (B18), the resulting equation
IS
(dQ)
C
4 u (Tg,x - Ta,x) dSa =
( dQ;; OE (Ta,x - Te4, dS,
X
Cancelling the common term dS, and dividing both sides of the
equation by U,
Tg,x - Ta,x = CJE (dQ),/(dQ;), [ 1 U
(Ti,x - $1 (I3201
Re-arranging (B20) so as to bring all terms involving Ta,x to the
left side of the equation,
T ax+" 9 Tg,x + OE (B21)
Equation (B21) is the general form of the expression for the local
armor temperature in terms of (dQ),/(dQz), and other entering
variables. Equation (30) of the Calculation Procedure is the same as
equation (B21). Although in equation (30) both dQ and dQi change
with x, the subscript x has been omitted from the ratio (dQ)/(dQE).
This has been done both to simplify the notation and to emphasize
that for the class of radiators studied, the ratio (dQ)/(dQg) is
independent of x and has the same value at every axial station
along the armor.
96
Derivation of the expression for S, x : The expression for the
surface exposed by the armor to space in the axial distance from
the radiator tube inlet to the station at x9 equation (34) of the
Calculation Procedure, is derived in this sub-section.
Differentiating equation (B20), treating e and U as constants,
transposing dT, x 9
to the right side of the eauation, and dropping
the subscript x with the understanding that only 0, E, U and T,
are not x-dependent,
dT g
= dTa+
From the heat balance for the gas,
- (rhc,)dT, = dQ
Using (~18) and (B19) for dQ,
Employing (B22) in (B241,
- hilt,) dT, + -!? d U
(dQ) =
(dQ;) 1
GE (Ta 4 4
- Te > 1 dS,
(B22)
(B23)
(B24)
--- -
97
Re-arranging this equation so that dS, appears by itself on the
left side of the equal sign, the expression for dSa is
iC ds, = - 2
(SE dTa
IilC CT,4 - 2) P I
(B25)
-- u
CT,4 - Te4) 1 Integrating dS, axially from the radiator tube entrance plane to
station x,
iC S P
a,x= -F dTa
IilC, - 2 In
u
r (dQ)
1 (dQ;) (Ta:x - Te4)
K
(T 4 a,en - Te4) sn
(B26)
Equation (B26) is the general form for Sa x in terms of the variable 9
(dQ)/(dQ;) l In the derivation of equation (B26), no restriction
has been placed on the manner in which (dQ)/(dQE) may vary; hence it
may vary in any desired manner, consistent with the overall thermal
and pressure performance required of the radiator. One possible
98
prescription is that (dQ)/(dQE) shall have the same value at all
axial stations of the radiator; another, less direct, but definitive
prescription is that (dQ)/(dQ;) shall vary so as to keep the thickness
of the external fins constant along the entire length of the radiator
tubes. Other specifications on (dQ)/(dQE) are also possible; each
specification leads to characteristic properties of the external fins.
For ease of calculation in the present study, (dQ)/(dQ{) was specified
to have a single constant value for the entire radiator6 Under this
specification, equation (B26) is reduced to the
r Ta,x
simple form
S 1 iC, dTa
a,x = - (SE (dQ)/(dQ;) /
Ta4 - Te4 T a,en
1
Performing the indicated integration, re-arranging the logarithmic
expression so as to obtain a positive algebraic sign, and writing
ic,/U in the equivalent form
(B27)
Ii, P -= SC,/ ~dQ)/(dQ;~
u - U /DdQ>/(dQ;fj '
99
equation (B27) becomes
S 1 1 T a,x =
I&, a,x + Te Ta,en - Te (dQ>/(dQg> 0~ 4T,3 + T a,en + Te Ta,x - Te
+ 2tan-1(T:r)- 2tan-l(';rn)] +
mc,/[(dQ)/(dQ;j
+ u /edQ,/(dQ;j
Equation (B28) is the same as equation (34) of the Calculation
Procedure.
Comments: In the foregoing discussion, the heat radiated by an element
of armor and its external fins, dQ, has been expressed in terms of the
heat radiated by an element of bare armor surface, dQt , as given by
equation (Bl9). It would have been possible to omit all references to
bare armor radiators and to postulate that the heat release of the
armor and its external fins, dQ, is expressible in the form
dQ = q(x) UE (T 4 89
- Te4)dSa 1 with V(X) a function of x whose form requires determination and
is governed by input specifications. It is instructive, however,
100
to retain the concept of a reference bare armor radiator, because
thereby the close relationships that exist between bare and finned
armor radiators are kept in view. For example, the data of Ref-
erences 2 and 3 show that the heat release of a finned armor
radiator is expressible conveniently and naturally as a multiple
of the heat release of a bare armor radiator.
A relationship between externally finned and externally bare armor
radiators of interest in the present study is as follows: The
class of externally finned radiators that operates with the same
value of (dQ)/(dQz) over the entire armor surface has fluid and armor
temperature fields, and surface area of armor, that are identical
with those of bare armor radiators which satisfy the conditions
T 0 = T, of finned armor radiator
'b = '
(T g,en)b = Tg,en
cTg,ex)b = Tg,ex
(dQ)/(dQ;) = constant
(Ihcp)b = lilcp
(dQ)/(dQ;)
U 11 u, =
(dQ)/(dQ;)
11
(dS,)b = dSa
(B29)
101
The condition (dSalb = dS, implies that equal increments of armor
surface are to be considered when comparing the axial progress of
T Ta' g' and dQ in the externally bare and externally finned armor
radiators.
The identity of gas and armor temperature fields, and of armor
surface areas, of bare and finned armor radiators that satisfy
(B29) is readily established by use of equations (B21), (B24) and
the condition (dS,)b = dS, . The relationship between bare and
finned armor radiators that satisfy (B29) can be used as a basis
for a calculation procedure which produces numerical results
identical with those reported herein.
102
REFERENCES
1. Glassman, A. J. : Summary of Brayton Cycle Analytical Studies for Space-Power System Applications. NASA TN D-2487, Sept. 1964.
2. Sparrow, E. M. and Eckert, E. R. G. : Radiant Interaction Between Fin and Base Surfaces. Feb. 1962.
Trans. ASME, Jnl of Heat Transfer,
3. Sparrow, E. M. ; Miller, G. B. ; and Jonsson, V. IS. : Radiating Effectiveness of Annular-Finned Space Radiators, Including Mutual Irradiation Between Radiator Elements. Jnl Aerosp. SC., Nov. 1962.
4. Loeffler, I. J. ; Lieblein, S. ; and Clough, N. : Meteoroid Protection for Space Radiators. NASA Paper 2543-62, prepared for ARS Space Power Systems Conf. at Santa Monica, Calif. , Sept. 25-28, 1962.
5. Sparrow, E. M. ; Jonsson, V. K. ; and Minkowycz, W. J. : Heat Transfer from Fin-Tube Radiators, Including Longitudinal Heat Conduction and Radiant Interchange Between Longitudinally Nonisothermal Finite Surfaces. NASA TN D-2077, Dec. 1963.
6. Humble, L. V. ; Lowdermilk, W. H. ; and Desmon, L. G. : Measure- ments of Average Heat-Transfer and Friction Coefficients for Subsonic Flow of Air in Smooth Tubes at High Surface and Fluid Temperatures. NACA Report 1020, 1951.
7. McAdams, W. H. : Heat Transmission. New York, 1954, p* 268.
McGraw-Hill Book Co.,
8. Diedrich, J. H. and Lieblein, S. : Materials Problems Associated With the Design of Radiators for Space Powerplants. Paper pre- pared for ARS Space Power Systems Conference, Santa Monica, Calif., Sept. 25-28, 1962.
9. DeLorenzo, B. ; and Anderson, E. D. : Heat Transfer and Pressure Drop of Liquids in Double-Pipe Fin-Tube Exchangers. Nov. 1945.
Trans. ASME,
10. Kays, W. M. ; and London, A. L. : Compact Heat Exchangers, McGraw-Hill Book Co., Inc., New York, 1958, fig. 84 and Table 20.
11. Svehla, R. A. : Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures. NASA TR R-132, 1962.
12. Saule, A. V. ; Krebs, R. P. ; and Auer, B. M. : Design Analysis and General Characteristics of Flat-Plate Central-Fin-Tube Sensible Heat Radiators. NASA TN D-2839, June, 1965.
13. Jakob, M. : Heat Transfer, Vol. I, John Wiley 8c Sons, Inc., New York, 1949, p. 132.
Alternator
f
--- Turbine
Heat source , -4 Recuperator 1,
-a ---- J I
Radiator
Compressor
U’ t !
---w------s- J
Figure 1. - Brayton cycle with gaseous working fluid in radiator.
104
/-Exhaust header /
r---------w---- _____
tubes PExternal fin
Externai
----) +Da fins A
2 A \ * .
I --- j 3n
n-u +-.. ----- -;f Armored tube
----- J ______ --,-L------
---
_--m-e- ______------
____----
L-S~~~ly header t- 8a
c 7 -7
(a) Radiator w .th unsplit headers (schematic).
LGas out I 6a
---a- --- 3 -r -------- -----
I 0-m -----, f I
.----.
J-
A ”
I V
-2eF-
t
External fin
Header branch
Armored tube
Gas in
(b) Rad .ator with once-split headers (schematic).
Figure 2. - Radiator and header arrangements.
105
0 (a) Internally unfinned
0.13" = b
tube.
Short radial fins - I Short radial fins - II
(b) Short radial fins. Fins are
Main Fins7\ continuou axially
IS
Lri-l (c) Radially long, axially continuous fins.
-Fins me sxi interrupted
CD-8609
(d) Radially long, axially interrupted fins.
Figure 3. - Principal internal geometries studied.
.ally
m--------m
m---D---.
---- H ---
-/ i
CD-8610 Figure 4. - Radiator tube with central external fins.
CD-8611
Figure 5. - Radiator tube with circumferential external fins.
108
Type fin Fig- Source ure of data
2x10-l
10-l
8
6
None Short radial Radially long,
axially continuous Radially long,
continuous fins
2x10-2
10-2
a L
10- 102 103 104 105
G deq Tav Refilm = - - pfilm Tfilm
Figure 6. - Heat transfer coefficients and friction factors.
Figure 7. - Fin length parameter versus (dQ)/(d<) and NcmIm for central external fins.
110
2.
2.
2.
2.
1.
1.
1.
1,
1
Figure a. - Illustrative theoretical performance curves for radiators with circumfer- ential external fins (ref. 3).
111
wall Exposed side of fin base
(a) Finned tube detail.
Side of fin base
(b) Fin symbols. CD-8612
Figure 9. - Geometric details and nomenclature of internally finned tubes.
q-Headers /'I
/ /
al fins
I I I 0
.92 .96 1.00 1.04 7 nR 1.12 1.16 L.“V
Tube inside diameter, in.
Figure 10. - Effect of tube inside diameter in internally unfinned radiators; Ztube = 25 ft, central external fins, Nc,~F = 1.0.
113
Planform area -
5OOd I
800
600
400
I I I I I I I I I
200
0 .2 .4 .6 .8 1.0
External fin conductance parameter, N,Q~
Figure 11. - Effect of Nc,~F in internally unfinned radiators; Ztube = 25 ft, central external fins, di = 1.07 inch, headers split once.
114
i ii 600 _-l
gwg 4-1 -- 7-- 9 Planform area
2 400 7 LI
1200
600
--- Unsplit headers
t I Each header split once
I I I I
1.60 1.64 1.68 1.72 1.76 Tube inside diameter, in.
Inteinal fins- I km- '.
c
1.80 1.84
Figure 12a. radial -
- Wfect of tube inside diameter in internally finned radiators; Mshort
Nc,LF
I" internal fins (Fig. 3b), +.ube = 25 ft, central external fins, = 1.0.
115
c I I I --- Unsplit headers
Each header nolit (see Fig. 2)
ante
a00
600
1.8 1.9 2.0 Tube inside diameter, in.
Figure 12b. - Effect of tube inside diameter in internally finned radiators; "short radial - II" internal fins (Fig. 3b), &be = 25 ft, central external fins, N ",$ = 1.0.
116
I I I I I I- O 10 Internal fins per tube, each header
split once --- Unsplit headers
Each header split once > (See Fig. 2)
\ \
A. . 1000 I I 1
-\ . \
\ . -- Total weight A-l4 I I I
A u-9
-.-I I /
800 "-<al weight.
'\
rTube walls (armor)
_^ 1
12oor
2.0 2.1 2.2 2.3 2.4 Tube inside diameter, in.
Figure 13. - Effect of tube inside diameter and of number of internal fins per tube; radially long, axially continuous internal fins (Fig. 3c), Ztube = 25 ft, central external fins, NC,+ = 1.0.
117
n = No. of int&nal fins per tube
4 400 8 12
/-
cl I 20
Planform area
600
Ekternal fins
2.4 2.6 2.8 3.0 Tube inside diameter, in.
3.2 3.4
Figure 14. - Effect of tube inside diameter and of number of internal fins per tube; axially interrupted fins (Fig. 3d), ztube = 25 ft, central external fins, N
=,kF = 1.0, each header split once (see Fig. 2).
118
400
n = No. of internal fins per tube
,.-- 20 I I I- /-
Planform area --I- ----+
rPlanform area de- creases by 25 ft2- if headers are
I I t.. I- split once
I I I I
=Tube wall weigh drops 20 lb and
-header weight d -175 lb, if head1
are split once ‘I T - nal fins
I
in
I-J
,rEkt'ernal ginsTiS
Internal fiyv /) A'20
0. .8 .9 1.0 1.1 1.2 1.3
Tube inside diameter, in.
I1 rl
t 7 rops ers
Figure 15. - Effect of tube inside diameter and of number of internal fins per tube; radially long, axially continuous internal fins (Fig. 3c), 'tube = 6 ft, central external fins, Nc,~ = 1.0, headers unsplit (see Fig. 2).
n = No. of internal fins per tube - 1 1 12 16
- 20 L
klanform area
looolll-l __I. 1 F I- 1--L
6oo\ 4,,i 2ooI
0 I& 1.1 1.2
I ’ Tube walls
1.3 1.6 Tube inside diameter, in.
Figure 16. - Effect of tube inside diameter and of number of internal fins per tube; axially interrupted fins (Fig. 3d), ltube = 6 ft, central external fins, N, L = 1.0, each header split once (see Fig. 2). 'F
I I I I
400
120
600
800
600
0 .2 .4 .6 .8 1.0
External fin conductance parameter, Nc,+
Figure 17. - Effect of external fin conductance parameter; I \ radially long, axially continuous internal fins (Fig. 3c), 2 - 25 ft, central external fins, tube inside dt%tk = 2.05 inch, 10 internal fins/tube, each header split once (see Fig. 2).
121
Tube walls
.2 .4 .6 .8 1.0 eternal fin conductance parameter, Nc,$,
Figure 18. - Effect of external fin conductance parameter; axially interrupted internal fins (Fig. 3d), Ztube = 25 ft, central external fins, tube inside diameter = 2.98 inch, 8 internal fins/tube, each header split once (see Fig. 2).
.2 .4 .6 .8 1.0 External fin conductance parameter, N, I,
'F Figure 19. - Effect of external fin conductance parameter;
radially long, axially continuous internal fins (Fig. 3d), kube = 6 ft, central external fins, tube inside diameter = 1.23 inch, 30 internal fins/tube, each header split once (see Fig. 2).
123
800
600
Planform area I I
fins per tube
II I I I I I I I I
01 I I I I I I I 12 InternA fins
I I
.2 .4 .6 .8 1.0 External fin conductance parameter, Nc,L~
Figure 20. - Effect of NC,% and of number of internal fins per tube; axially interrupted internal fins (Fig. 3d), Ztube = 6 ft, central external fins, each header split once (see Fig. 2).
124
4wJ
200 E
160
120
h Tube i.d. I 1
---t -- I I I
n = No. of interrA fins per tube
-6
anform ea
I 12
I
4
I I I . -Eased on curves-
.Ol .02 .03 .04 External fin conductance parameter, Nc,Ra
Figure 21. - Effect of Nc,R, and of number of internal fins per tube; circumferential external fins, Ro/R, = 4. Each header split once (see Fig
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